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Transformation of Coordinates Between Two Horizontal Geodetic Transformation of coordinates between two horizontal geodetic datums by Petr Vaní ek Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton, NB, Canada, E3B 5A3 and Robin R. Steeves Microsearch Corporation PO Box 59, Juniper, NB, Canada, E0J 1P0 Abstract. The following topics are discussed in this paper: the geocentric coordinate system and its different realizations used in geodetic practice; the definition of a horizontal geodetic datum (reference ellipsoid) and its positioning and orientation with respect to the geocentric coordinate system; positions on a horizontal datum and errors inherent in the process of positioning; and distortions of geodetic networks referred to a horizontal datum. The problem of determining transformation parameters between a horizontal datum and the geocentric coordinate system from known positions is then analysed. It is often found necessary to transform positions from one horizontal datum to another. These transformations are normally accomplished through the geocentric coordinate system and they include the transformation parameters of the two datums as well as the representation of the respective network distortions. Problems encountered in putting these transformations together are pointed out. Geocentric Coordinate Systems and their Realizations By definition, a geocentric coordinate system is a system whose origin (0, 0, 0) coincides with the centre of mass, C, of the Earth, and whose axes are fixed by convention. There are infinitely many such coordinate systems differing from each other by the orientation of their axes. The most common geocentric system used in geodesy is the Conventional Terrestrial (CT) system which is oriented so that the z-axis points towards the Conventional International Origin (CIO), the x-axis lies in the Conventional Greenwich Meridian, and the y-axis makes, with the other two axes, a right-handed Cartesian triad [Vaní ek and Krakiwsky, 1986]. This system provides the most convenient link with geodetic astronomy. Two other geocentric coordinate systems, the Instantaneous Terrestrial, referred to the instantaneous spin axis of the Earth, and the Natural Geocentric system, whose axes coincide with the Earth’s principal axes of inertia, provide similarly convenient links with astronomical observations and with the dynamics of the Earth respectively. Positions in the CT-system are sometimes given in Cartesian coordinates (x,y,z), and sometimes in curvilinear geodetic coordinates ( φ, λ, h) i.e., in geodetic latitude, longitude, and height. The use of curvilinear geodetic coordinates, however, requires the introduction of a geocentric reference ellipsoid (co- axial with the Cartesian system) with major semi-axis a and minor semi-axis b. The geodetic height h, sometimes called the ellipsoidal height, is the height of a point above the surface of that reference ellipsoid. In some literature, the biaxial reference ellipsoid is called the reference spheroid. Curvilinear geodetic coordinates are easily transformed into Cartesian coordinates by the following formula: , (1) where N denotes the radius of curvature of the ellipsoid in the direction perpendicular to the meridian plane [Vaní ek and Krakiwsky, 1986]. The inverse transformation has to be solved either iteratively, through a linearization, using an algebraic equation of fourth degree [Vaní ek and Krakiwsky, 1986], or some other techniques. The Geodetic Reference System of 1980 (GRS 80), recommended for use in geodesy by the International Association of Geodesy (IAG) in 1980 [IAG, 1980], utilizes the CT-system combined with a reference ellipsoid deemed to closely fit the actual shape of the Earth. It is given by major-semi axis a = 6 378 137 m and flattening (2) equal approximately to 1/298·25. It is now used almost universally in geodetic works. A practical realization of GRS 80 is the well known World Geodetic System of 1984 (WGS 84) [Defense Mapping Agency, 1987]. Points can be positioned directly in this coordinate system either through orbits of positioning satellite systems (Transit or GPS) or by positioning relative to some already existing points. The North American Datum of 1983 (NAD 83) is another, continent-wide realization of GRS 80. Geodetic Coordinate Systems - Their Positioning and Orientation Before the advent of positioning satellite systems, it had not been possible to realize and use geocentric coordinate systems. Geodetic (G) coordinate systems [Vaní ek and Krakiwsky, 1986], with reference ellipsoids selected to fit the shape of the Earth (more rigorously, the shape of the Earth’s gravity equipotential surfaces) the best in a regional manner, were and still are used in most countries. More than 150 of such G-systems exist. A biaxial reference ellipsoid associated with a G-system (in the same way that the geocentric reference ellipsoid is associated with the CT-system) is called a Geodetic Horizontal Datum . A geodetic horizontal datum is non-geocentric, i.e., the centre E of the ellipsoid is displaced from the Earth’s centre of mass C, usually by hundreds of metres. As well, the axes of a geodetic horizontal datum (the axes of the G-system), are misaligned with respect to the CT-system. The misalignments, however, are small, mostly less than one second. We note that the term "geodetic horizontal datum" is sometimes used for the conglomerate of the reference ellipsoid and a geodetic network (see below) referred to it. This usage can be misleading and should not be encouraged. To be of practical use in positioning, a G-system, and thus even its geodetic horizontal datum, must be positioned with respect to the Earth. Prior to the advent of satellite positioning, when geocentric positioning was not possible, the only way of positioning and orientating horizontal datums had been to do it with respect to the Local Astronomical (LA) coordinate system of a selected point. (The LA-system is defined by the local gravity vertical and the spin axis of the Earth [Vaní ek and Krakiwsky, 1986].) Six defining parameters had to be chosen at the Initial Point ( also called the Datum Point) : geodetic latitude, geodetic longitude, geoidal height, two components of the deflection of the vertical, and the geodetic azimuth of a line originating at this point [Vaní ek and Krakiwsky, 1986]. In the case of some prior information being available, the values of the geodetic latitude and longitude of the Initial Point were chosen so as to achieve the best fit between astronomic and geodetic latitudes and longitudes at a number of deflection points. Such was the case with NAD 27 and ED 1950. This approach was designed to ensure that the reference ellipsoid would fit the geoid in the region of interest. As the adjustment was carried out in two dimensions, the result was equivalent to the case described above except that the defining parameters would implicitely refer to another point, that point being the centroid of the deflection points used. With the advent of satellite positioning (with respect to the CT-system), it became possible to indirectly position and orient a datum with respect to the CT-system by comparing coordinates of the same points in the G-system and the CT-system. This technique has been used, for example, to realize the CT-system as NAD 83, and is, in fact, the only technique available to position and orientate an existing horizontal datum with respect to the CT-system. A direct conversion of the six defining parameters into position and orientation parameters (with respect to the CT-system) is not possible; the vital link, the position and orientation of the LA-system at the Initial Point with respect to the CT-system, is missing. We know only the directions of the LA-axes in the G-system (through Φ, Λ, and A-α) while the geocentric position of the Initial Point is not known at all. Before we turn to the transformation between the G-system and the CT-system, let us discuss some aspects of position determination on a horizontal datum. These aspects will be of importance in the transformation. Geodetic Horizontal Positions Positions (coordinates) of various points referred to a geodetic horizontal datum have been determined by geodesists and surveyors for over several hundred years. The original use of these points was in mapping, and their positions, specified by latitude φ and longitude λ on the particular datum used, are essentially horizontal positions even though heights h above the datum might have been known for some of the points. Let us discuss this statement in some detail. Horizontal positions have been, and are being, determined by relative positioning (determining coordinates of new points by making measurements from points with known coordinates). Relative positioning by terrestrial means consists of measuring distances, angles, and azimuths. Relative positioning by spatial means consists of making satellite or VLBI measurements. Thus the coordinates of one point and the direction from one point to another, or the coordinates of several points, must be known to start the process of positioning. A horizontal datum is therefore of no practical use unless at least one position and a direction is known beforehand, to which the relative positions may be tied. In the classical case, the initial point and the azimuth from that point to another point serve this purpose. In practice, relative horizontal positioning is applied repeatedly, resulting in networks of points whose horizontal positions are determined simultaneously in one adjustment computation. The height h of a "horizontal" network point is normally determined (e.g. by measuring vertical angles) to an accuracy suitable for computing corrections needed to reduce terrestrial observations from the Earth’s surface, where they are made, to the reference ellipsoid, where they are needed for position computations. Because levelling is not normally used to determine orthometric heights H of network points, and geoidal heights N were known to a relatively poor accuracy, the geodetic heights were usually determined to an accuracy of about ten times lower than horizontal positions, and should therefore not be mixed with latitude and longitude to create a three-dimensional position.
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